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t t/s2

Non-Boussinesq model

Fig. 5.167 Four conceptual stages of a two-layer ocean during the process of flushing, simulated in terms of the Boussinesq and non-Boussinesq models.

Fig. 5.167 Four conceptual stages of a two-layer ocean during the process of flushing, simulated in terms of the Boussinesq and non-Boussinesq models.

increase from stage 2 to stage 3 is

Obviously such a state is gravitationally unstable, and overturning takes place. With the heavy layer sinking to the bottom, from stage 3 to stage 4 the total GPE is reduced

Evolution of GPE during this process is shown in Figure 5.168. Note that GPE in stage 4 is slightly higher than in the initial stage 1,

X4 - X1 = 0.5gh2p0$ (S2 - Si) = 0.5gh2p0a (T2 - Tx) > 0 (5.295)

If a non-Boussinesq model is used, the total mass of each layer remains unchanged, but its thickness shrinks during cooling. Thus, from stage 2 to stage 3, the upper layer thickness is reduced to h' ~ [1 - a(T2 - Ti)]h (5.296)

Owing to the downward movement of the center of mass of the upper layer, the total GPE is reduced

This stage is gravitationally unstable, and overturning takes place, during which the total GPE is reduced

Layer flipping

Energy

Layer flipping

Energy

Time

Non-Boussinesq model

Time

Non-Boussinesq model

Fig. 5.168 Evolution of GPE in a two-layer ocean during the process of flushing, as simulated in terms of Boussinesq and non-Boussinesq models; assume that a = ß = 1 and T, S, and energy are in nondimensional units.

The evolution of GPE for the non-Boussinesq model is also shown in the lower part of Figure 5.168.

It is clear that only the non-Boussinesq model can capture the evolution of GPE accurately. In fact, GPE of the system is reduced during the steps shown in Figure 5.168. At the end of these steps, GPE of the system is at its lowest. To return to the initial stage, the cold water in the lower layer has to be warmed up. Expansion of the lower layer pushes its center, and thus the center of the layer above, upward.

On the other hand, GPE in the Boussinesq model is falsely increased during the transition from stage 2 to stage 3. Furthermore, the final stage 4 in the Boussinesq model has slightly more GPE than the initial stage; another artifact in the model due to the artificial source of mass introduced during cooling. Finally, to return to the initial stage 1, the model must go through the heating of the lower layer. During heating of the lower layer, an artificial sink of mass takes place, resulting in a reduced GPE of the lower layer, while the GPE of the upper layer remains unchanged. Therefore, only in the transition from stage 3 to stage 4 does the change in GPE stay the same in Boussinesq and non-Boussinesq models, since this transition does not involve a change in density of the water in both layers.

The flushing phenomenon also exists for the case with wind stress forcing. The period of flushing depends on the parameters of the model, but it is always on the order of thousands of years. The cycle of flushing can be clearly seen from many indexes of the circulation, such as the meridional overturning rate (Fig. 5.169a), the sea surface salinity (Fig. 5.169b), the basin mean temperature (Fig. 5.169c), and the heat loss (Fig. 5.169d) (Huang, 1994).

Fig. 5.169 Time evolution of thermohaline circulation in a single-hemisphere Boussinesq model ocean forced by wind stress, thermal relaxation, and freshwater flux, including: a meridional overturning rate (in Sv); b mean sea surface salinity (deviation from the mean salinity of 35); c basin-mean temperature (in °C); d heat flux to atmosphere (not counting the heat flux from atmosphere to the ocean, in PW) (Huang, 1994).

Fig. 5.169 Time evolution of thermohaline circulation in a single-hemisphere Boussinesq model ocean forced by wind stress, thermal relaxation, and freshwater flux, including: a meridional overturning rate (in Sv); b mean sea surface salinity (deviation from the mean salinity of 35); c basin-mean temperature (in °C); d heat flux to atmosphere (not counting the heat flux from atmosphere to the ocean, in PW) (Huang, 1994).

During flushing, meridional overturning is extremely strong, and relatively warm and salty water is brought to the upper ocean, making the sea surface salinity higher. At the same time, rapid heat loss to the atmosphere reduces the basin mean temperature, as shown in Figure 5.169c, d.

An interesting application of this idea is the deep ocean anoxia during the Late Permian era (Zhang et al., 2001). As the models, including a three-box model and an oceanic general circulation model, have indicated, the oceanic circulation at that time may have switched between a long-time haline mode and a short burst of thermal mode, with a period of approximately 3,330 years.

During the phase of the thermal mode (Fig. 5.170), there may exist a superposition of the wind-driven circulation and the thermohaline circulation looping through the Tethys Sea. The meridional overturning cell is strong, with sinking near the South Pole, and it penetrates all the way to the sea floor (Fig. 5.170b). The circulation is much faster, with a large meridional heat flux, and is southward at all latitudes.

In comparison, during the phase of the haline mode, the meridional circulation is much slower and confined to the upper 1.5 km (Fig. 5.171). The meridional heat flux is much smaller and more symmetrical with respect to the equator.

5.5 Combining wind-driven and thermohaline circulation

5.5.1 Scaling of pycnocline and thermohaline circulation

The wind-driven circulation in the upper ocean can be clearly described in terms of the horizontal advection of water within the Ekman layer and the horizontal gyre below. We

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